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We study the topology of line singularities, which are complex hypersurface germs with non-isolated singularity given by a smooth curve. We describe the degeneration of its Milnor fiber to the singular hypersurface by means of a vanishing polyhedron in the Milnor fiber. As a milestone, we also study the topology of the degeneration of a complex isolated singularity hypersurface under a nonlocal point of view.

We describe all rigid algebras and all irreducible components in the variety of four-dimensional Leibniz algebras ${𝔏𝔢𝔦𝔟}_4$ over $ℂ.$ In particular, we prove that the Grunewald–O'Halloran conjecture is not valid and the Vergne conjecture is valid for ${𝔏𝔢𝔦𝔟}_4.$

The goal of this work is to study geometric properties of geometrically irreducible subschemes on degenerations of Fano varieties (more generally, of separably rationally connected varieties). It is known that these geometrically irreducible subschemes exist when the ground field has characteristic zero or contains an algebraically closed subfield. We show that the dimension of this geometrically irreducible subscheme has a lower bound by the Fano index of the generic fiber.

The degradation degree prediction of lithium-ion battery has been studied through experimental data. Characterization parameters on the degradation degree of lithium-ion battery were deduced under consideration of the internal and external factors. The analysis of discrete degree was proposed to depict the degradation degree for lithium-ion battery. Furthermore, based on fuzzy inference system (FIS), the predicted model of the degradation degree for lithium-ion battery was built and its output was defined as the degenerate coefficient $\beta$, $\beta \in [0,1]$. Finally, by learning, training and simulating, the FIS model has been validated to be reliable and applicable in prediction on the degradation degree of lithium-ion battery. The simulation results show that the degradation degree of lithium-ion battery is more serious when $\beta$ is closer to 1, and the degradation degree is lighter when $\beta$ is closer to 0.

Some characteristic parameters were epurated in this paper by analyzing internal and external factors of the degradation degree of lithium-ion battery. These characteristic parameters include open circuit voltage (OCV), state of charge (SOC) and ambient temperature. The degradation degree was evaluated by discrete degree of the array, which is composed of the above parameters. The epurated parameters were verified through adaptive neuro-fuzzy inference system (ANFIS) model building. The expression of degradation coefficient was finally determined. The simulation results show that the expression is reasonable and precise to describe the degradation degree.

A finite poset (P ≼ S) determines a finite dimensional algebra T_P over the field 𝔽 of two elements, with an upper triangular representation. We determine the structure of the radical of the representation algebra A of the monoid (T_P,·) over a field of characteristic different from 2. We also consider degenerations of A over a complete discrete valuation ring with residue field of characteristic 2.

We give a complete algebraic classification of nilpotent binary Lie algebras of dimension at most 6 over an arbitrary field of characteristic not 2 and a complete geometric classification of nilpotent binary Lie algebras of dimension 6 over $ℂ$. As an application, we give an algebraic and geometric classification of nilpotent anticommutative $ℭ𝔇$-algebras of dimension at most 6.

In this paper we prove that the Kähler–Einstein metrics for a degeneration family of Kähler manifolds with ample canonical bundles converge in the sense of Cheeger–Gromov to the complete Kähler–Einstein metric on the smooth part of the central fiber when the central fiber has only normal crossing singularities inside smooth total space. We also prove the incompleteness of the Weil–Peterson metric in this case.

In this paper, we obtain several structure results for a class of spherical subgroups of connected reductive complex algebraic groups that extends the class of strongly solvable spherical subgroups. Based on these results, we construct certain one-parameter degenerations of the Lie algebras corresponding to such subgroups. As an application, we exhibit explicit algorithms for computing the set of spherical roots of such a spherical subgroup.

In an earlier paper we defined a relation ≤_Δ between objects of the derived category of bounded complexes of modules over a finite dimensional algebra over an algebraically closed field. This relation was shown to be equivalent to the topologically defined degeneration order in a certain space for derived categories. This space was defined as a natural generalization of varieties for modules. We remark that this relation ≤_Δ is defined for any triangulated category and show that under some finiteness assumptions on the triangulated category ≤_Δ is always a partial order.

We give a geometric classification of complex $4$-dimensional nilpotent $ℭ𝔇$-algebras. The corresponding geometric variety has dimension 18 and decomposes into $2$ irreducible components determined by the Zariski closures of a two-parameter family of algebras and a four-parameter family of algebras. In particular, there are no rigid $4$-dimensional complex nilpotent $ℭ𝔇$-algebras.

We give algebraic and geometric classifications of complex four-dimensional nilpotent noncommutative Jordan algebras. Specifically, we find that, up to isomorphism, there are only $18$ non-isomorphic nontrivial nilpotent noncommutative Jordan algebras. The corresponding geometric variety is determined by the Zariski closure of three rigid algebras and two one-parametric families of algebras.

The aim of this work is to discuss the concepts of degeneration, deformation and rigidity of Hopf algebras and to apply them to the geometric study of the varieties of Hopf algebras. The main result is the description of the n-dimensional rigid Hopf algebras and the irreducible components for n<14 and n = p² with p a prime number.

Three different notions of geometric degeneration for finite dimensional algebras are compared. It is proved that for algebras A₀ and A₁ of same dimension, if A₀ is a CB-degeneration of A₁, then A₀ is a degeneration of A₁ in the classical sense; moreover, if additionally A₀ and A₁ are basic, then A₀ is a CB-degeneration of A₁ if and only if A₀ is a rigid degeneration of A₁. It is also shown that CB-degenerations along affine lines in the different dimension case do not differ so much from those in the equal dimension case.

This survey is based on my lectures given in the last few years. As a reference, constructions of moduli spaces of parabolic sheaves and generalized parabolic sheaves are provided. By a refinement of the proof of vanishing theorems, we show, without using vanishing theorems, a new observation that \dim H^0({𝒰}_C,\Theta {𝒰}_C) is independent of all of the choices for any smooth curves. The estimate of various codimensions and computation of canonical line bundle of moduli space of generalized parabolic sheaves on a reducible curve are provided in Section 6, which is completely new.

We describe all degenerations of the variety ${\mathfrak{J}\mathfrak{o}\mathfrak{r}\mathfrak{d}}_3$ of Jordan algebras of dimension three over $\mathbb{C}$. In particular, we describe all irreducible components in ${\mathfrak{J}\mathfrak{o}\mathfrak{r}\mathfrak{d}}_3$. For every $n$ we define an $n$-dimensional rigid “marginal” Jordan algebra of level one. Moreover, we discuss marginal algebras in associative, alternative, left alternative, non-commutative Jordan, Leibniz and anticommutative cases.

In this paper, we consider the Galois covers of algebraic surfaces of degree 6, with all associated planar degenerations. We compute the fundamental groups of those Galois covers, using their degeneration. We show that for 8 types of degenerations, the fundamental group of the Galois cover is non-trivial and for 20 types it is trivial. Moreover, we compute the Chern numbers of all the surfaces with this type of degeneration and prove that the signatures of all their Galois covers are negative. We formulate a conjecture regarding the structure of the fundamental groups of the Galois covers based on our findings.

Degeneration of the intervertebral disc (IVD) and associated spinal disorders are a leading cause of morbidity resulting in substantial pain and increased health cost. Current treatment options may relieve some clinical symptoms but will never stop further degeneration in the IVD. In recent years, several investigations have been carried out trying to regenerate IVD tissue. These biomedical engineering methods aim to retard or even reverse the degenerative process and possibly regain a healthy IVD. In this review, recent progresses on each category of biomedical engineering approaches will be briefly introduced.

A reproducible experimental model of cervical spondylosis in rodents was established by means of detachment of the back paravertebral muscles from the vertebrae and resection of the spinous processes together with supraspinous and interspinous ligaments. Spinal instability elicited by this surgical intervention accelerated the process of intervertebral disc degeneration. It induced cervical spondylosis in mice when extended over a 6 — 12 month period. The pathologic changes in the cervical intervertebral discs of this animal model were progressive: proliferation of cartilaginous tissue and fissures in the annulus fibrosus; shrinkage or disappearance of the nucleus pulposus; and herniation of disc materials and osteophyte formation.

Basic fibroblast growth factor (bFGF) was reported to stimulate proliferation and matrix synthesis of cultured intervertebral disc cells. Using this animal model, the distributions of cells expressing bFGF and FGF receptor (FGF-R) were investigated in normal and degenerating intervertebral discs by immunohistochemistry and in situ hybridization histochemistry. Expression of bFGF protein and FGF-R messenger RNA were found in actively proliferating chondrocytes in the degenerating annulus fibrosus, whereas these were negative in normal annulus fibrosus. This suggested that bFGF might regulate the process of disc degeneration.

The availability of this experimental model should be valuable for further understanding, both biochemically and biomechanically, of the pathogenesis of cervical spondylosis.